Extending Existing Dependency Theory to Temporal Databases∗ Christian S. Jensen Richard T. Snodgrass Michael D. Soo Abstract Normal forms play a central role in the design of relational databases. Several normal forms for temporal relational databases have been pro- posed. These definitions are particular to specific temporal data models, which are numerous and incompatible. This paper attempts to rectify this situation. We define a consis- tent framework of temporal equivalents of the important conventional database design concepts: functional dependencies, primary keys, and third and Boyce-Codd normal forms. This framework is enabled by mak- ing a clear distinction between the logical concept of a temporal relation and its physical representation. As a result, the role played by temporal normal forms during temporal database design closely parallels that of normal forms during conventional database design. These new normal forms apply equally well to all temporal data models that have timeslice operators, including those employing tuple timestamping, backlogs, and attribute value timestamping. As a basis for our research, we conduct a thorough examination of ex- isting proposals for temporal dependencies, keys, and normal forms. To demonstrate the generality of our approach, we outline how normal forms and dependency theory can also be applied to spatial and spatiotemporal databases. ∗The authors have the following addresses and affiliations. C. S. Jensen is with Aalborg University, Datalogi, Fredrik Bajers Vej 7E, DK{9220 Aalborg Ø, Denmark, [email protected]. R.T.Snodgrass and M. D. Soo are with Department of Computer Science, University of Arizona, Tucson, AZ 85721, USA, rts,soo @cs.arizona.edu. f g 1 1 Introduction A central goal of relational database design is to produce a database schema, consisting of a set of relation schemas. Each relation schema is a collection of attribute names and their associated domains. Normal forms are an attempt to characterize \good" relation schemes. A wide variety of normal forms has been proposed, the most prominent being third normal form and Boyce-Codd normal form. An extensive theory has been developed to provide a solid formal footing. There is also a need for temporal normal forms and underlying con- cepts that may serve as important guidelines during temporal database design. In response to this need, an array of temporal normalization concepts have been previously proposed, including first temporal nor- mal form [Segev & Shoshani 88], two different variants of time normal form [Ben-Zvi 82, Navathe & Ahmed 89], and P and Q normal forms [Lorentzos & Kollias 89]. The proposals are significant since each, in the context of a particular temporal data model, can be used to design temporal database schemas. However, the specificity of the proposals is a weakness since a given normal form inherits the inherent peculiarities of its data model, and, having chosen a particular temporal normal form, it is unsatisfactory to be required to define all of the normal forms anew for each of the two dozen existing temporal data models [Snodgrass 92], should another model be better suited for representing the semantics of the application. Furthermore, the existing normal forms often deviate substantially in nature from conventional normal forms and are in some sense not \true" extensions of these, for a variety of reasons that we detail later in this paper. In this paper, we show how temporal normal forms, including the related concepts of temporal dependencies and temporal keys, may be defined so that they apply to all temporal data models, and so that temporal database design concepts closely parallel their conventional counterparts. We do not simply focus on a single temporal data model. Instead, we utilize a new data model, termed the bitemporal conceptual data model (BCDM), that is, in some sense, the \largest common denom- inator" of existing temporal models [Jensen et al. 94B]. Specifically, we have shown how to map relations and operations in several quite dif- ferent temporal relational data models into relations and operations in 2 this data model. This is an important property, as it ensures that the normal forms expressed in this model are applicable also to other mod- els. We define the temporal normal forms in the context of this model. Our proposal accommodates valid-time, transaction-time and bitempo- ral relations [Snodgrass & Ahn 86, Jensen et al. 94A]. We also note that the BCDM has been adopted as the underlying data model of the con- sensus temporal query language TSQL2 [Snodgrass et al. 94]. Design of TSQL2 schemas thus directly benefits from the definitions of temporal dependencies and normal forms introduced here. Our focus is on the design of temporal database schemas. A sub- stantial body of work exists on the specification and efficient checking of more general temporal, or dynamic, integrity constraints. The depen- dencies, keys and normal forms of this paper can be seen as constraints on database instances, but are different in two respects. Unlike general constraints, they impact the design of temporal databases. And since the focus is on database design, on-line checking of the constraints is not of relevance here. We also believe that additional normalization concepts are needed that take the different temporal characteristics of data into considera- tion, but the development of such concepts is beyond the scope of this paper. Instead, this paper is restricted to providing data-model inde- pendent mappings of the existing conventional normalization concepts to temporal databases. We also limit the scope of the paper to so-called intra-state depen- dencies [B¨ohlen94]. Intra-state dependencies are defined in terms of individual snapshots of a temporal database. For example, the con- ventional notions of functional dependency and multivalued dependency are, by definition, intra-state dependencies. Inter-state dependencies, on the other hand, express constraints between attribute values in different snapshots. The paper is organized as follows. In Sections 2, 3, and 4, we examine all existing definitions, to our knowledge, of temporal dependencies, keys, and normal forms, respectively. This is the first thorough survey of work in these areas. Each section first briefly describes the relevant conventional normalization concepts, and lists a number of important properties that should carry over to their temporal counterparts. On this basis, the temporal database design proposals known to us are introduced and evaluated. The existing definitions satisfy many, but not all, of 3 the properties required of entirely natural extensions of conventional normal forms. The existing temporal design concepts provide a valuable foundation upon which we subsequently build. The topic of Section 5 is the bitemporal conceptual data model. We describe the type of relation supported by the BCDM, and briefly de- scribe a few algebraic operators needed to support the new temporal normal forms defined in the next section, where we then develop tem- poral counterparts of the conventional dependencies, keys, and normal forms, again, limiting ourselves in this paper to intra-state variants. This is done is such a way that virtually all of the conventional normaliza- tion theory carries over to the temporal context. The result is that the role played by temporal normal forms during temporal database design closely parallels that of normal forms during conventional database de- sign. This is possible, in part, because of a careful choice of temporal data model. To demonstrate the generality of our approach, we outline, in Sec- tion 8, how normal forms and dependency theory can also be applied to spatial and spatiotemporal databases [Al-Taha et al. 93]. Conclusions and future research are the subject of Section 9. 2 Previous Proposals for Temporal Depen- dencies In this section, we consider previous proposals of temporal dependencies (in chronological order, of course!). To provide a basis for this, we first review the definition of the conventional functional dependency, and then we highlight those properties that we feel should also be satisfied by corresponding temporal database dependencies. 2.1 Conventional Functional Dependency Throughout the paper, we generally use R to denote an arbitrary rela- tion schema, and r(R) to denote that r is an instance of R. Explicit (non-temporal) attributes of a relation schema are generally denoted A1;:::An, and X and Y are used to denote sets of attributes. For tuples, the symbol s is used (possibly indexed), and s[X] denotes the projection of tuple s onto the attributes X. 4 For the purpose of database design, a functional dependency [Codd 72A] is an intensional property of a database schema. Definition: Let relation schema R be defined as R = (A1;A2;:::;An), and let X and Y be sets of attributes of R. The set Y is functionally dependent on the set X, denoted X Y , if for all meaningful instances r of R ! s1; s2 r (s1[X] = s2[X] s1[Y ] = s2[Y ]): 8 2 ) If X Y , we say that X determines Y . A functional dependency X Y!is trivial if Y X. ! ⊆ ut A functional dependency constrains the set of possible extensions of a relation. Which functional dependencies are applicable to a schema reflects the reality being modeled and the intended use of the database. Determining the relevant functional dependencies is a primary task of the database designer. The two most important normal forms, third normal form [Codd 72] and Boyce-Codd normal form [Codd 74], as well as the concept of key, all rely on the concept of functional dependency. Example: To illustrate, consider a database recording the phone num- bers, departments, and employees in a company. This can be modeled with the schema Emp = (Name; Dept; PhNo). In this company, an em- ployee can belong to only one department, meaning that Name Dept. An employee may have several phone numbers, so Name does not! deter- mine PhNo.